$\DeclareMathOperator{\sep}{sep}$Consider Spaces which are $T_1$ but may fail to be $T_2$. For $x,y\in X$ define $x\leq y$ iff $\sep(x) \subseteq \sep(y)$ where $\sep(x)$ refers to the set of all $y$ such that $\{x,y\}$ is a $T_2$ subspace.
Q: Is this preorder respected by continuous maps? By inverse image?